A nonlinear model reduction based on eigenmode decomposition and projection for the prediction of sub- and supercritical limit-cycle oscillation is presented herein. The paper focuses on the derivation of the reduced-order model formulation to include expansion terms up to fifth order such that higher-order nonlinear behaviour of a physical system can be captured. The method is ...

In this paper the \(N-1\) nonlinear modal interactions that occur in a nonlinear three-degree-of-freedom lumped mass system, where \(N=3\), are considered. The nonlinearity comes from springs with weakly nonlinear cubic terms. Here, the case where all the natural frequencies of the underlying linear system are close (i.e. \(\omega _{n1}: \omega _{n2}:\omega _{n3} \approx 1:1:1\)) ...

The motion planning problem for nonholonomic robotic systems is studied using the continuation method and the optimization paradigms. A new Jacobian motion planning algorithm is derived, based on a solution of the Lagrange-type optimization problem addressed in the linear approximation of the system. Performance of the new algorithm is illustrated by numeric computations performed ...

Many modern applications of the flexible multibody systems require formulations that can effectively solve problems that include large displacements and deformations having the ability to model nonlinear materials. One method that allows dealing with such systems is continuum-based absolute nodal coordinate formulation (ANCF). The objective of this study is to formulate an ...

The dynamic response of a nonlinear system with three degrees of freedom in resonance that is loaded, inter alia, with a non-ideal excitation is investigated. A direct current motor (DC motor) with an eccentrically mounted rotor serves as a non-ideal source of energy. The general coordinate corresponding to the rotor dynamics steadily increases as a result of rotational motion. The ...

In this study, the quaternion Mandelbrot sets and Julia sets (abbreviated as M–J sets) with additive and multiplicative noise perturbations are constructed and the changes of their fractal characteristics are explored. The experimental results show that the quaternion M sets with additive noise perturbations move in the direction of the noise, while the structures remain unchanged. ...

This paper brings a new mathematical model of the third-order autonomous deterministic dynamical system with associated chaotic motion. Its unique property lies in the existence of circular equilibrium which was not, by referring to the best knowledge of the authors, so far reported. Both mathematical analysis and circuitry implementation of the corresponding differential equations ...

Nonlinear effects in vibration responses are investigated for the undamaged composite plate and the composite plate with a delamination. The analysis is focused on higher harmonic generation in vibration responses for various excitation amplitude levels. This effect is investigated using the triple correlation technique. The dynamics of composite plate was modelled using ...

Many models of three-dimensional rigid body dynamics employ Euler parameters as rotational coordinates. Since the four Euler parameters are not independent, one has to consider the quaternion constraint in the equations of motion. This is usually done by the Lagrange multiplier technique. In the present paper, various forms of the rotational equations of motion will be derived, and ...

In many practical situations, it is impossible to measure the individual trajectories generated by an unknown chaotic system, but we can observe the evolution of probability density functions generated by such a system. The paper proposes for the first time a matrix-based approach to solve the generalized inverse Frobenius–Perron problem, that is, to reconstruct an unknown ...

We present chaotic dynamics of flexible curvilinear shallow Euler–Bernoulli beams. The continuous problem is reduced to the Cauchy problem by the finite-difference method of the second-order accuracy and finite element method (FEM). The Cauchy problem is solved through the fourth- and sixth-order Runge–Kutta methods with respect to time. This preserves reliability of the obtained ...

In this paper, we consider cooperative manipulation of a planar rigid body using multiple actuator agents—unilateral thrusters, each attached to the body and each able to apply an unilateral force to the body. Generally, the dynamics of the body manipulated with uncoordinated forces of thrusters is nonlinear. The problem we consider is how to design the unilateral force each agent ...

Time delays widely exist in the complex dynamical systems. In this paper, time-delay complex dynamical systems are studied by analyzing the properties of corresponding Julia sets. Experimental results show that the stability of complex dynamical system changes greatly due to the influence of time delay. The variations in time-delay Julia sets are studied. In addition, the ...

Controlling system dynamics with use of the Largest Lyapunov Exponent (LLE) is employed in many different areas of the scientific research. Thus, there is still need to elaborate fast and simple methods of LLE calculation. This article is the second part of the one presented in Dabrowski (Nonlinear Dyn 67:283–291, 2012). It develops method LLEDP of the LLE estimation and shows that ...

This paper presents a control strategy designed as a combination of a PD controller and a twisting-like algorithm to stabilize the damped cart pole system, provided that the pendulum is initially placed within the upper-half plane. To develop the strategy, the original system is transformed into a four-order chain of integrator form, where the damping force is included through an ...

This paper proposes an improvement in cross-correlation methods derived from the Lee–Schetzen method, in order to obtain a lower mean square error in the output for a wider range of the input variances. In particular, each Wiener kernel is identified with a different input variance and new formulas for conversion from Wiener to Volterra representation are presented.